please help me with math

The required expressions are ¬ ∃x(F(x, Ralph)), ∀xF(x, Fred), and  ∀x∃yF(x,y). Here, ¬ means negation, ∃ means existential quantifier, and ∀ means universal quantifier.

One or more variables that are defined on a particular domain are expressed by a predicate. By giving the variable a value or by quantifying it, it is possible to turn a predicate with variables into a proposition.

There are two quantifiers used in this they are (i) existential quantifier which is represented by ∃ and (ii) universal quantifier which is represented by ∀.

For, the proposition, "Nobody trusts Ralph", the quantifier is written as      ¬ ∃x(F(x, Ralph)).

For the proposition, "Everybody trusts Fred", the quantifier is written as ∀xF(x, Fred).

For the proposition, "Somebody trust everybody", the quantifier is written as ∀x∃yF(x,y).

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The conclusion about given functions are given below.

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).

We have the following theorem:

If f is a quadratic function of the form f(x) = ax²+bx+c and ac < 0 , then the function f has two x-intercepts.

(a) g(x) = -8x²+5x-2

For this function a = -8 and c = -2, then -8 × -2 = 16 is greater than zero. Therefore, we cannot conclude anything.

(b) h(x) = [tex]-\frac{1}{3}x^{2} + 3x[/tex]

For this function a = -8 and we don't know the value of c. Therefore, we cannot conclude anything.

(c) k(x) = 8x²-5x-7

For this function a = 8 and c = -7, then 8 × -7 = -56 is less than zero. Therefore, we can conclude that the function k has two x-intercepts.

(d) j(x) = [tex]-\frac{71}{99}x^{2} + 210[/tex]

For this function a = [tex]-\frac{71}{99}[/tex] and c = 210, then [tex]-\frac{71}{99}[/tex] × 210 = [tex]-\frac{4970}{33}[/tex] is less than zero. Therefore, we can conclude that the function k has two x-intercepts.

(e) f(x) = 4x²-3x+7

For this function a = 4 and c = 7, then 4×7 = 28 is greater than zero. Therefore, we cannot conclude anything.

(f) F(x) = [tex]-x^{4} + x^{3} + 9[/tex]

We cannot conclude anything because this is not a quadratic function.

The conclusion about given functions are given above in explanation.

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complete question : Following is a statement of a theorem which can be proven using the quadratic formula. For this theorem, a, b, and c are real numbers. Theorem If f is a quadratic function of the form f .x/ D ax2 C bx Cc and ac < 0, then the function f has two x-intercepts. Using only this theorem, what can be concluded about the functions given by the following formulas(a) g (x) = -8x^2 + 5x - 2 (b) h (x) = -(1/3)x^2 + 3x (c) k (x) = 8x^2 - 5x - 7 (d) j (x) = -(71/99)x^2 + 210 (e) f (x) = 4x^2 - 3x + 7 (f) F (x) = -x^4 + x^3 + 9

Answer:

a.

b. 35/100 of 76 = 26 to 27 remained

c.
d. statistic

e. experiment

f. I don't think so

Step-by-step explanation:

A graph that represents the solution set of the compound inequality is: graph D.

In Mathematics, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the following inequality symbols:

Next, we would solve the given compound inequality by making x the subject of formula as follows;

x + 3 < 1/2(4x - 12) < 20

Multiplying all through by 2, we have the following:

2x + 6 < 4x - 12 < 40

Next, we would solve the compound inequality in parts:

2x + 6 < 4x - 12

4x - 2x < 12 + 6

2x < 18

x < 18/2

x < 9.

4x - 12 < 40

4x < 40 + 12

4x < 52

x < 52/4

x < 13.

In conclusion, the line on a number line (graph) should be open when the inequality symbol is greater than (>) or less than (<).

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The fact that some people who smoked never develop lung cancer does not disprove the evidence.

The fact that some people who have never smoked develop lung cancer does not disprove the evidence linking smoking to lung cancer.

Smoking is a major risk factor for lung cancer, and the overwhelming majority of people who are diagnosed with the disease are smokers or former smokers. However, it is important to note that other factors, such as exposure to radon and other pollutants, can also increase the risk of lung cancer, and that some people may be more genetically susceptible to the disease. So, it is possible for people who have never smoked to develop lung cancer as well.

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Internet of Things refers to the technology that allows data, collected from sensors in all types of machines, to be sent over the Internet to repositories where it can be stored and analyzed.

Internet of Things refers to the collective network of connected devices and the technology that facilitates communication between devices and the cloud, as well as between the devices themselves.

There are some Advantages of IOT:

It can assist in the smarter control of homes and cities via mobile phones. It enhances security and offers personal protection.

Automating activities saves us a lot of time. Information is easily accessible, even if we are far away from our actual location, and it is updated frequently in real time.

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When σ is known, hypothesis tests are performed for one population mean. The one-mean z-test, or simply z-test, is a hypothesis test carried out for a single population mean when the population standard deviation is known.

A data distribution's spread is measured by standard deviation. It calculates the average separation between each data point and the mean.

Whether we consider the data to be a population unto itself or a sample of a broader population affects the method we use to calculate standard deviation.

Statistics is the parent topic for the Population Mean notion. In a nutshell, statistics addresses the issues of gathering data as well as its analysis, interpretation, and presentation. The population mean formula calculates the average of all the values in the population or set of data.

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Answer:

12 times y = x

Step-by-step explanation:

I see on you paper you want to do intervals of 2 so just take 12 and multiply it by the desired number. Equation: 12 x # = the y axis point. This is where #=x-axis and plugging it in the equation gives you the y point.

It can be concluded that in 2018, approximately 21.7% of children and adolescents in the United States displayed serious psychological disturbances and needed clinical treatment.

The percentage of children and adolescents in the United States displaying serious psychological disturbances is calculated by dividing the number of children and adolescents with serious psychological disturbances by the total number of children and adolescents in the United States. This can be expressed mathematically as:

% = (Number of children and adolescents with serious psychological disturbances / Total number of children and adolescents in US) x 100

Using estimates from 2018, approximately 48.5 million children and adolescents in the United States, and 10.5 million of them displaying serious psychological disturbances, the percentage of children and adolescents in the United States displaying serious psychological disturbances in 2018 can be calculated as:

% = (10.5 million / 48.5 million) x 100 = 21.7%

Therefore, it can be concluded that in 2018, approximately 21.7% of children and adolescents in the United States displayed serious psychological disturbances and needed clinical treatment.

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When we are aligning two shafts, the reason behind rotating the coupling hubs before taking each measurement is so that each of the measurements is taken from the same place on the hub every time.

Shafts are basically a component of circular cross section which rotates and transmits the power from a driving device like an engine or a motor through a particular machine. Coupling hubs are basically the attachment of the coupling to shaft in a device.

During the alignment of two shafts, the angular as well as parallel misalignment is thoroughly checked. The coupling hubs are rotated before taking each measurement is so that each of the measurements is taken from the same place on the hub every time.

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Answer:

Step-by-step explanation:

You want the number of calories in each dish if 1 portion of hamburger and fries with 2 portions of fettuccine Alfredo has 4240 calories, and 2 portions of hamburger and fries with 1 portion of fettuccine Alfredo has 3980 calories.

The equations describing the calories in each dish can be written using h and f for the calories in a portion of hamburger and in a portion of fettuccine, respectively:

  1h + 2f = 4240

  2h +1f = 3980

Subtracting the first equation from twice the second gives ...

  2(2h +f) -(h +2f) = 2(3980) -(4240)

  3h = 3710 . . . . . . . simplify

  h = 1240 . . . . . . . divide by 3

Then the calories in the fettuccine can be found from the second equation:

  f = 3980 -2h = 3980 -2480 = 1500

The actual number of calories in a portion of hamburger and fries is 1240.

The actual number of calories in a portion of fettuccine Alfredo is 1500.

__

Additional comment

The attachment shows a calculator solution using matrix methods.

Answer:

The Z-score (also known as a standard score) represents the number of standard deviations a value is away from the mean of a distribution. The formula for the Z-score is:

Z = (X - μ) / σ

Where X is the value of interest, μ is the mean of the distribution, and σ is the standard deviation of the distribution.

Given the information in the problem, we know that the mean of the fruit fly's lifespan is 32 days and the standard deviation is 5 days.

To find the Z-score of a fruit fly whose lifespan is 37 days:

Z = (37 - 32) / 5 = 1

To find the Z-score of a fruit fly whose lifespan is 30 days:

Z = (30 - 32) / 5 = -0.4

To find the Z-score of a fruit fly whose lifespan is 47 days:

Z = (47 - 32) / 5 = 2.4

Therefore, a fruit fly whose lifespan is 37 days has a Z-score of 1, a fruit fly whose lifespan is 30 days has a Z-score of -0.4, and a fruit fly whose lifespan is 47 days has a Z-score of 2.4.

As per the permutation method, the number of possible way for here travel is 5.

The term permutation in math is known as an arrangement of objects in a definite order and the members or elements of sets are arranged here in a sequence or linear order.

Here we have given that Amanda got a train ticket valid for the five cities listed below.

There are Amsterdam, brussels, Lille, Paris, Rotterdam.

So, the given number of ways is 5.

Then the value of n = 5 and the value of r = 1 because here we have given that she can choose to visit some or none of these cities which means that she must chose at least one city.

Then the number of ways is calculated as,

=> ⁵P₁ =  5!/ (5 - 1)!

=> ⁵P₁ = 5!/4!

=> ⁵P₁ = 5

Complete Question:

Let us consider that Amanda got a train ticket valid for the five cities listed below. she can choose to visit some or none of these cities. she doesn't have time to visit all of them.

They are Amsterdam, brussels, Lille, Paris, Rotterdam

Find out the number of ways to visit each cites using permutation method.

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Answer:

A system of linear equations is a set of two or more equations containing two or more variables. The solution to a system of linear equations is the set of values for the variables that makes all the equations in the system true.

For example, a system of linear equations whose solution is (6,19) could be represented by the following equations:

x + 2y = 25

3x - 4y = 6

Solving this system would yield x = 6 and y = 19 as the solution.

If cement costs $76 per cubic yard, the total cost of cement for a sidewalk with a volume of 480 inches³ is $1,013.

The total cost of cement for the sidewalk is determined using unit conversions and multiplication.

After determining the volume in inches and converting it to yards, the result is multiplied by the unit cost of cement per cubic yard.

Dimensions of the Sidewalk:

Length = 40 inches

Width = 3 inches

Thickness = 4 inches

Volume = 480 inches³ (40 x 3 x 4)

36 Inches = 1 Yard

480 cubic inches = 13.33 cubic yards

Cost of cement per cubic yard = $76

Cost of the sidewalk = $1,013 ($76 x 13.33)

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Qualitative data is data that cannot be counted easily using numbers is not have the correct definitions.

Qualitative data is used to address the research questionnaire regarding whats and how many(s) or also for research procedures to quantify a problem. This data can be used to compare or count data on a numerical scale.

This type of research typically involves in-depth interviews, focus groups, or observations in order to collect data that is rich in detail and context. This data is used for Qualitative research which is very helpful when understanding social reality and the attributes, beliefs, and motivations of people.

The main importance of this kind of project come into the limelight when someone is trying to understand the reason behind one's beliefs or behaviors.

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Answer: Quadrilateral

Step-by-step explanation:

hexagon: 6 sides

octagon: 8 sides

quadrilateral: 4 sides

triangle: 3 sides

The quadrilateral is the only one that makes sense.

In a case whereby you mix 1 half cup of salt with 3 cups of sugar the ratio of salt to sugar.

The concept that will be used is ratio. An ordered pair of numbers a and b, represented as a / b, is a ratio if b is not equal to 0. A proportion is an equation that sets two ratios at the same value. For instance, you could express the ratio as follows: 1: 3 if there is 1 boy and 3 girls (for every one boy there are 3 girls)

we were told that there were 1 half cup of salt with 3 cups of sugar

Then the ratio can be written as

salt/sugar

= (1 1/2)/3

Since the amount of the salt is in mixed fraction, we can reduce to improper fraction for easy calculation.

=( 3/2)/ 3

= 3/6

=1/2

Hence, the ratio is 1:2

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missing options:

A. 2:3

B.1:2

C.2:1

D. 5:6

The presented system of linear equations has no solution.

The final form of the augmented matrix suggests that the system of linear equations has no solution, as the last row represents an equation with 0 = 0, which is always true but doesn't provide any information about the variables.

To see this more clearly, let's use the augmented matrix to write out the system of linear equations:

1x - 1y + 2z = 4

2y + 1z = -1

z = 1

The first two equations are consistent and have a unique solution, but the last equation is trivial and provides no information about x and y. In other words, infinitely many pairs of x and y satisfy the first two equations, but none satisfy the third equation. Therefore, the given system of linear equations has no solution.

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The rational function in the simplest form of (f/g)(x)  is x / 5x - 2.

Composite functions are when the output of one function is used as the input of another. A typical example of a composite function is (f/g)(x).

Therefore, let's solve the rational function in the simplest form.

Hence,

f(x) = -x

g(x) = -5x + 2

(f/g)(x) = f(x) / g(x)

Therefore,

(f/g)(x) = - x / -5x + 2

Hence,

(f/g)(x) = - x / - (5x - 2)

Finally,

(f/g)(x) = x / 5x - 2

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The weight of the coalition P1, P2, P3 is 36

From the question, we have the following parameters that can be used in our computation:

weighted voting system [20: 9,7,5,3,1]

The variable P1, P2 and P3 are the first three values

So, we have

P1 = 20

P2 = 9

P3 = 7

The weight of the coalition P1, P2, P3 is the sum of the points

So, we have

Coalition = 20 + 9 + 7

Coalition = 36

Hence, the weight coalition is 36

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The complete table is given as follows -

Brain (part)    0.6    1.3    1.8     3

Body (part)    6.6   14.3  19.8   33

Total (whole) 7.2   15.6  21.6   36

Given is that an interior designer recently spent £1359.55 buying 25 kettles and 20 toasters for a smaller group of new houses. For her current assignment she needs 80 kettles and 56 toasters.

The given table is -

Brain (part)   0.6   B   1.8      E

Body (part)   A    14.3   19.8  33

Total (whole) 7.2  C      D     36

We can write the expressions as -

0.6 + A = 7.2                                           .... Eq( 1 )

A = 6.6

1.8 + 19.8 = D                                            .... Eq( 2 )

21.6 = D

E + 33 = 36

E = 3                                                              .... Eq( 3 )

Therefore, the complete table is given as follows -

Brain (part)    0.6    1.3    1.8     3

Body (part)    6.6   14.3  19.8   33

Total (whole) 7.2   15.6  21.6   36

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The factors and zeros of the function f(x)=x2-5x+6 are x-3, x-2 and 3 and 2 respectively

Given the quadratic function as;

f(x) = x² - 5x + 6

Using the factorization method of solving quadratic function, we have to multiply the coefficient of x² by the constant of the function

Then find the pair factors of the function that sums up to -5

The pair factors area -2 and -3

Substitute the values, we get;

x²- 2x - 3x + 6

Pair the expression

(x² -2x) - (3x + 6)

Factorize the value

x(x -2) -3(x - 2)

The zeros are;

x-3 = 0

x = 3

Then,

x -2 = 0

x =2

Hence, the values are 2 and 3

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The difference between 4.507 and 4.517 can be expressed as -0.010.

The concept that will be used here is substraction. The outcome of subtracting one integer from another is difference. While we frequently discuss differences in how objects appear, feel, or even taste, mathematics demonstrates how much two numbers vary from one another in terms of quantity. Difference is what is left behind after subtracting one number from another.

the outcome of subtraction of two numbers. the difference between two numbers. For instance: There is a 5 difference between 8 and 3.

We were to find the difference between 4.507 and 4.517 whichg implies the substraction between the values and can be done as

(4.507 - 4.517)= -0.010

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complete question:

What is the difference between 4.507 and 4.517

Circumference of circle = 7π yards

The correct option is B.

A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance (radius) from a fixed point (center) on the plane. The fixed point is called the origin or center of the circle and the fixed distance of the points from the origin is called the radius.

Given,

Segment drawn from one point on the circle through the center to another point = diameter of circle = 7 yards

Circumference of the circle = π × Diameter

                                              = 7π yards

Hence, 7π yards is the circumference of the circle

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Answer:

b

Step-by-step explanation:

The direction and position of the fourth and fifth arrows indicate that the object then moves towards the origin, with a smaller velocity.

Motion maps are used to illustrate the direction and position of an object. Using the motion map, the description of the object position and velocity is as follows:

The object starts its movement from the origin with a large velocity, before moving back to the origin with a smaller velocity. It stops for 1 second in the origin, then moves away with a larger velocity, Finally, it moves back towards the origin with a smaller velocity.

See attachment for the motion map, where the number on each arrow in the map, represents the position of the object.

Note that; the long arrow means large velocity while the short arrow means small velocity.

We analyze the direction and position using the arrows

Therefore, the direction and position of the fourth and fifth arrows indicate that the object then moves towards the origin, with a smaller velocity.

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Answer:

Based on the motion map, the object starts at the reference point "x" on the left side of the position line. The three dots with short vector arrows moving toward "x" indicate that the object is moving in the direction of the reference point and gaining speed. This suggests that the object is accelerating. Then, there is a single dot, which might indicate a momentary pause or a constant velocity. Afterward, the two dots with longer vector arrows moving away from "x" suggest that the object is decelerating or slowing down. In summary, the motion map shows an object that starts from rest, accelerates towards the reference point, maintains a constant velocity for a brief moment, and then decelerates away from the reference point.

Step-by-step explanation:

Use chat gpt to give you unique answer.

The population of the town at any given time t can be determined using the formulas provided. For example, if the town's population at time t=5 years is desired, the population can be calculated as follows:(a) P(5) = 1600 + 70(5) = 2100 people (b) P(5) = 1600(1.09)^5 = 2266.05 people

(a) P(t) = 1600 + 70t

(b) P(t) = 1600(1.09)^t

(a) In this case, the population increases by 70 people per year, so the formula for the population as a function of time is P(t) = 1600 + 70t. This means that the population is 1600 people at time t=0 and increases by 70 people for each additional year.

(b) In this case, the population increases by 9 percent each year, so the formula for the population as a function of time is P(t) = 1600(1.09)^t. This means that the population is 1600 people at time t=0 and increases by a factor of 1.09 (9 percent) for each additional year.

The population of the town at any given time t can be determined using the formulas provided. For example, if the town's population at time t=5 years is desired, the population can be calculated as follows:

(a) P(5) = 1600 + 70(5) = 2100 people

(b) P(5) = 1600(1.09)^5 = 2266.05 people

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If k is the number of classes and n is the number of observation then for class intervals, select the smallest k such that 2k >= n using statistics.

n = 36

2^5 = 32< 36

2^6 = 64 > 36

So k = 6

a) Frequency distribution: Answer A.

Class interval                 Frequency

0 to less than 1000        11

1000 to less than 2000        9

2000 to less than 3000 3

3000 to less than 4000 3

4000 to less than 5000 5

5000 to less than 6000 5

b) Relative frequency distribution: Answer B.

Class interval                        Relative frequency

0 to less than 1000            11 / 36 = 0.306

1000 to less than 2000            9 / 36 = 0.250

2000 to less than 3000    3 / 36 = 0.083

3000 to less than 4000     3 / 36 = 0.083

4000 to less than 5000     5 / 36 = 0.139

5000 to less than 6000     5 / 36 = 0.139

c) Cumulative relative frequency:

Class interval         Cumulative Frequency

less than 1000         11

less than 2000         20

less than 3000         23

less than 4000         26

less than 5000          31

less than 6000          36

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The probability of event a occurring is 0 and the conditional probability of event b occurring given that event a has occurred is 0.5.

The probability that both event A and event B occur is calculated using the formula P(A and B) = P(A) x P(B|A).

In this case, P(A) = 0 and P(B|A) = 0.5. When we plug these values into the equation, P(A and B) = 0 x 0.5 = 0. Therefore, the probability of both event A and event B occurring is 0. The probability that both events a and b occur is determined by multiplying the probability of event a occurring with the conditional probability of event b occurring given that event a has already occurred. This can be expressed mathematically as P(A∩B)=P(A)*P(B|A). In this case, the probability of both events a and b occurring is 0.5*0.5=0.25, since the probability of event a occurring is 0 and the conditional probability of event b occurring given that event a has occurred is 0.5.

This makes sense, since event A has a probability of 0 and cannot occur in the first place, so the probability of both events occurring together is 0.

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It is verified that the given vectors x₁ and x₂ are solutions of the given system. Using Wronskian it is shown that they are linearly independent. Then, [tex]W(t)=7e^{-3t}[/tex]. The general solution of the given system is written as [tex]x(t)=\left[\begin{array}{c}3c_1e^{2t}+c_2e^{-5t}\\2c_1e^{2t}+3c_2e^{-5t}\end{array}\right][/tex].

Wronskian analysis helps to determine whether a solution is linearly dependent or independent.

Given the system is [tex]x'=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x[/tex].

Let's differentiate x₁ concerning t, and we get,

[tex]\begin{aligned}x_1'&=\left[\begin{array}{c}\frac{d}{dt}(3e^{2t})&\\\frac{d}{dt}(2e^{2t})&\end{array}\right] \\&=\left[\begin{array}{c}6e^{2t}&\\4e^{2t}&\end{array}\right] \end{aligned}[/tex]

Now,

[tex]\begin{aligned}\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x_1&=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]\left[\begin{array}{c}3e^{2t}\\2e^{2t}\end{array}\right]\\&=\left[\begin{array}{c}12e^{2t}-6e^{2t}\\18e^{2t}-14e^{2t}\end{array}\right]\\&=\left[\begin{array}{c}6e^{2t}\\4e^{2t}\end{array}\right]\\&=x_1'\end{aligned}[/tex]

From this, we can write,

[tex]x_1'=\left[\begin{array}{cc}4&-3\\6&-7\end{array}\right]x_1[/tex]

Therefore, we conclude that x₁ is a solution to the given system.

Let's differentiate x₂ concerning t, and we get,

[tex]\begin{aligned}x_2'&=\left[\begin{array}{c}\frac{d}{dt}(e^{-5t})&\\\frac{d}{dt}(3e^{-5t})&\end{array}\right] \\&=\left[\begin{array}{c}-5e^{-5t}&\\-15e^{-5t}&\end{array}\right] \end{aligned}[/tex]

Now,

[tex]\begin{aligned}\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x_2&=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]\left[\begin{array}{c}e^{-5t}\\3e^{-5t}\end{array}\right]\\&=\left[\begin{array}{c}4e^{-5t}-9e^{-5t}\\6e^{-5t}-21e^{-5t}\end{array}\right]\\&=\left[\begin{array}{c}-5e^{-5t}\\-15e^{-5t}\end{array}\right]\\&=x_2'\end{aligned}[/tex]

From this, we can write,

[tex]x_2'=\left[\begin{array}{cc}4&-3\\6&-7\end{array}\right]x_2[/tex]

Therefore, we conclude that x₂ is also a solution to the given system.

Now, find the Wronskian of x₁ and x₂, we get,

[tex]\begin{aligned}W(t)&=\text{det}[x_1\;x_2]\\&=\text{det}\left[\begin{array}{cc}3e^{2t}&e^{-5t}\\2e^{2t}&3e^{-5t}\end{array}\right] \\&=(3e^{2t}\times3e^{-5t})-(e^{-5t}\times2e^{2t})\\&=9e^{2t-5t}-2e^{2t-5t}\\&=9e^{-3t}-2e^{-3t}\\&=7e^{-3t}\\&\neq0\end{aligned}[/tex]

From this, we can conclude that x₁ and x₂ are independent.

Finally, we write the general solution of the system as follows,

[tex]\begin{aligned}x(t)&=c_1x_1+c_2x_2\\&=c_1 \left[\begin{array}{c}3e^{2t}\\2e^{2t}\end{array}\right] +c_2\left[\begin{array}{c}e^{-5t}\\3e^{-5t}\end{array}\right] \\&=\left[\begin{array}{c}3c_1e^{2t}\\2c_1e^{2t}\end{array}\right] +\left[\begin{array}{c}c_2e^{-5t}\\3c_2e^{-5t}\end{array}\right]\\&=\left[\begin{array}{c}3c_1e^{2t}+c_2e^{-5t}\\2c_1e^{2t}+3c_2e^{-5t}\end{array}\right] \end{aligned}[/tex]

The complete question is -

First, verify that the given vectors are solutions of the given system. Then use the Wronskian to show that they are linearly independent. Finally, write the general solution of the system.

[tex]x'=\left[\begin{array}{ccc}4&-3\\6&-7\end{array}\right]x;\; x_1=\left[\begin{array}{ccc}3e^{2t}\\2e^{2t}\end{array}\right], \;x_2=\left[\begin{array}{cc}e^{-5t}\\3e^{-5t}\end{array}\right][/tex]

To know more about Wronskian:

https://brainly.com/question/14309076

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